On relative bounds for interacting Fermion operators

We consider a system of interacting fermions whose Hamiltonian is unitarily transformed so that the interaction is a quartic perturbation of the Hartree-Fock effective Hamiltonian. It is shown under natural model assumptions that the interaction does not admit a relative bound with respect to the effective Hamiltonian that is uniform in the system's size. This bound is exemplified on the Hubbard model with nearest neighbor interaction on a discrete d-dimensional torus of length L around its Hartree-Fock ground state and derive relative bounds of the effective interaction with respect to the effective kinetic energy. It is shown that there are no relative bounds uniform in L.